Fast and Accurate Computation of Gauss-Legendre and Gauss-Jacobi Quadrature Nodes and Weights

@article{Hale2013FastAA,
  title={Fast and Accurate Computation of Gauss-Legendre and Gauss-Jacobi Quadrature Nodes and Weights},
  author={Nicholas Hale and Alex Townsend},
  journal={SIAM J. Sci. Comput.},
  year={2013},
  volume={35}
}
An efficient algorithm for the accurate computation of Gauss--Legendre and Gauss--Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton's root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The $n$-point quadrature rule is computed in $\mathcal{O}(n)$ operations to an accuracy of essentially double precision for any $n\geq 100$. 

Iteration-Free Computation of Gauss-Legendre Quadrature Nodes and Weights

  • I. Bogaert
  • Computer Science
    SIAM J. Sci. Comput.
  • 2014
TLDR
A series expansion for the zeros of the Legendre polynomials is constructed and a series expansion useful for the computation of the Gauss-Legendre weights is derived, providing a practical and fast iteration-free method to compute individual Gaussian quadrature node-weight pairs in O(1) complexity and with double precision accuracy.

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  • D. Brzezinski
  • Computer Science
    2018 Federated Conference on Computer Science and Information Systems (FedCSIS)
  • 2018
TLDR
The results of numerical experiments presented in the paper prove high accuracy and efficiency of developed methods for computation of quadratures' nodes and weights, decreased amount of required iterations for polynomials zeros finding and elimination of truncation errors during weights computation.

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Numerical evidence is provided showing that for degrees greater than 100 the asymptotic methods are enough for a double precision accuracy computation of the nodes and weights of the Gauss–Hermite and Gauss-Laguerre quadratures.

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

TLDR
Numerical evidence is provided showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation of the nodes and weights of the Gauss–Hermite and Gauss-Laguerre quadratures.

Asymptotic expansions of Jacobi polynomials and of the nodes and weights of Gauss-Jacobi quadrature for large degree and parameters in terms of elementary functions

  • A. Gil
  • Mathematics, Computer Science
  • 2020
TLDR
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...

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